4.1 Maximum and Minimum Values 4.1 Maximum and Minimum Values Definition: • A function f has an absolute maximum (or global maximum) at c if f(c) ≥ f(x) for all x in D, where D is the domain of f. The number f(c) is called maximum value of f on D. • Similarly, f has an absolute minimum at c if f(c) ≤ f(x) for all x in D and the number f(c) is called the minimum value of f on D. • The maximum and minimum values of f are called the extreme values of f. Local Maximum and Minimum Definition: • A function f has a local maximum (or relative maximum) at c if f(c) ≥ f(x) when x is near c. • Similarly, f has a local minimum at c if f(c) ≤ f(x) when x is near c. Example: Absolute maximum (also local maximum) Local maximum Local minimum Local minimum Absolute minimum (also local minimum) Extreme Value Theorem: If f is continuous over a closed interval, then f has absolute maximum and minimum over that interval. Maximum & minimum at interior points Maximum & minimum at endpoints Maximum at interior point, minimum at endpoint Example when a continuous function doesn’t have minimum or maximum because the interval is not closed. 4 No Maximum 3 2 1 yx 2 D 0, 2 -2 -1 0 1 2 No Minimum Suppose we know that extreme values exist. How to find them? Absolute maximum (also local maximum) Local maximum Local minimum Notice that local extremes in the interior of the function occur where f is zero or f is undefined. Fermat’s Theorem If f has a local maximum or minimum at c, and if f ′(c) exists, then f ′(c) = 0 . Note: When f ′(c) = 0 , f doesn’t necessarily have a maximum or minimum at c. (In other words, the converse of Fermat’s Theorem is false in general). Example when f ′(c) = 0 but f has no maximum or minimum at c . 2 yx 3 1 -2 -1 0 1 2 f0 -1 -2 (not an extreme) Critical numbers • Definition: A critical number of a function f is a number c in the domain of f such that either f ′(c) = 0 or f ′(c) doesn’t exist. • Example: Find the critical numbers of f(x) = x1/2(x-3) Solution: x 3 x 3 2 x 3x 3 f ( x) 2 x x 2 x 2 x Thus, the critical numbers are 0 and 1. • If f has a local maximum or minimum at c , then c is a critical number of f . The closed interval method for finding absolute maximum or minimum To find the absolute maximum and minimum values of a continuous function f on a closed interval [a,b]: 1. Find the values of f at the critical numbers of f in (a,b) . 2. Find the values of f at the endpoints of the interval. 3. The largest of the values from Steps 1 and 2 is the absolute maximum value; the smallest of these values is the absolute minimum value. Examples on the board.